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Theorem disjss2 3996
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3174 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2618 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 2964 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A
y  e.  B ) )
42, 3syl 15 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A y  e.  B
) )
54alimdv 1607 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A
y  e.  C  ->  A. y E* x  e.  A y  e.  B
) )
6 df-disj 3994 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x  e.  A
y  e.  C )
7 df-disj 3994 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
85, 6, 73imtr4g 261 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    e. wcel 1684   A.wral 2543   E*wrmo 2546    C_ wss 3152  Disj wdisj 3993
This theorem is referenced by:  disjeq2  3997  0disj  4016  uniioombllem2  18938  uniioombllem4  18941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-rmo 2551  df-in 3159  df-ss 3166  df-disj 3994
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